I am performing a t-test on the same couple of vectors in R. The first vector is the list of daily returns on a stock. The second vector is the list of daily log - returns on the same stock. Now, I understand the assumptions why we do a paired t-test as opposed to a Welch t-test (the default t-test setting in R). These are following results:
Welch t-test
t.test(dailyreturns, logReturn)
Welch Two Sample t-test
data: dailyreturns and logReturn
t = 0.10813, df = 2350, p-value = 0.9139
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-0.0006801156 0.0007595007
sample estimates:
mean of x mean of y
0.0005027479 0.0004630553
Paired t-test
t.test(dailyreturns, logReturn, paired = TRUE)
Paired t-test
data: dailyreturns and logReturn
t = 15.866, df = 1175, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
3.478409e-05 4.460108e-05
sample estimates:
mean of the differences
3.969258e-05
My question is given that the alternate hypothesis in both tests is the same and as the author mentions (see answer to Problem 14), while the assumptions in the Welch t-test is incorrect, the results are same. However, given the vastly p-values in both test, the null-hypothesis in one test will be rejected in the Welch t-test but won't be rejected in the paired t-test! And the graph in Problem 12 (from the same data set) shows that there isn't much difference in daily returns and daily log-returns. Should it not mean that the p-value should show insignificant results and therefore we will fail to reject the null hypothesis that the true difference in means is equal to 0? (Here, I am assuming true means ~ 0 and not EXACTLY 0 - am I right in this assumption?).
